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Global-in-time classical solutions for the 3D relativistic Vlasov–Maxwell Cauchy problem

Establish global-in-time existence of classical solutions to the three-dimensional relativistic Vlasov–Maxwell system for general initial data, beyond the special cases of lower spatial dimensions, cylindrical symmetry, or small initial data regimes in which global results are known.

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Background

The relativistic Vlasov–Maxwell system models the dynamics of a collisionless plasma where charged particles interact self-consistently with electromagnetic fields. While global weak solutions were constructed by DiPerna and Lions, the question of global-in-time existence for classical (smooth) solutions in three spatial dimensions remains unresolved in general.

Existing positive results cover lower-dimensional settings, symmetry-restricted configurations (e.g., cylindrical symmetry), or small initial data, for which global existence and even large-time behavior have been established. Extending these results to general three-dimensional data without such restrictions is a longstanding challenge.

References

For what concerns classical solutions, the global-in-time 3D Cauchy problem is still up to now a well-known open problem; see e.g., [23, 13, 37, 44, 40] for the classical conditional results and recent advances, except for the case of lower dimensions [20, 21], with cylindrical symmetry [49], and for small initial data [24, 22, 11, 50, 51, 12]; in the latter, the large time behavior of solutions was also established.

— Linear Landau damping for the Vlasov-Maxwell system in $\mathbb{R}^3$ (2402.11402 - Han-Kwan et al., 17 Feb 2024) in Section 1, Introduction